Document SE-METHODS · v1.4 · June 2026

The Methods Behind The Forge

Every formula. Every source. Every assumption we bake in. By Sal Misseri.

The archery industry ships calculators with brand logos on them and no math underneath. Black boxes. I built The Forge differently because I wanted a workshop, not a calculator. This page is that engine: the formulas, the assumptions, the tradeoffs, the bugs I've found, and the things I built that didn't exist anywhere else before The Forge shipped.

The Forge has a public version history and a privacy & data contract. If you find something wrong, write to sal@sparrowexpeditions.com.

§ 1FOC — the Sparrow Hybrid Model

Five tiers calibrated to modern 70-lb compound equipment at 280+ fps: Low (under 8%), Standard (8–11%), Recommended hunter range (11–15%), High (15–19%), Extreme (19%+). The 11–15% band is the practical sweet spot for most North American compound hunters.

Pressure-tested against the PCA 2026 Arrow Ballistics Study (Yates et al.). Their regression shows broadhead group radius keeps tightening with FOC; The Sparrow framework explains why — above ~15% the trajectory tax and spine sensitivity start to outweigh the accuracy gain for most hunting use cases.

FOC formula: FOC % = ((balance_point − L/2) / L) × 100 per the Easton ATA standard.

§ 2Energy & momentum

KE [ft·lb] = (W_grains × V_fps²) / 450 240
p  [slug·ft/s] = (W_grains × V_fps) / 225 400

Easton's canonical constants. Game-class thresholds (25 / 42 / 65 ft·lb) are policy classifications, not derivations — built from the Easton Hunting Reference Card and Pope & Young / state DNR recommendations.

§ 3Drag & ballistic coefficient

Empirical velocity-loss-per-yard linear approximation collapsing the textbook drag equation. Serviceable for archery velocities and ranges. BC formula BC = (W/7000)/(i·d²) with form factors i = 0.95 (field), 1.30 (fixed broadhead), 1.10 (mechanical).

§ 4Wind drift

McCoy's lag-time formula (Modern Exterior Ballistics, §7) with a per-vane drift-coefficient layer measured in the PCA 2025 study at 120 yd. The relative ordering reproduces PCA's results exactly. Firenock airfoil-tier coefficients (AV2/AV3) are engineering estimates flagged pca: false in the data table.

§ 5Trajectory & sight tape

Midpoint-rule integration of the velocity field. drop = ½ g t² with g = 32.174 ft/s². Similar-triangles pin geometry. Cross-checked against OnTarget!2.

§ 6Velocity prediction (IBO + Podium offsets)

Standard ATA IBO baseline with per-bow chrono offsets from Josh Jones (Podium MFJJ)'s 38-bow chronograph series. Median offset ⋅ ⋅ 11 fps; ±5 fps confidence for a known bow versus ±15 fps for an unmodeled one.

§ 7Dynamic spine

Stu Miller's Dynamic Spine model, cross-calibrated against the Easton 2024 Hunting Shaft Selector. Surfaced as an estimate, not a substitute for paper / bareshaft / broadhead tuning. Calibrated within ±15 spine units in the 250–500 range.

§ 8Atmospheric corrections

Pressure-altitude / ISA temperature correction at 66.7 ft per °F. Barometric formula ρ/ρ₀ = (1 − k·DA)^4.2561 from NACA Report 1135. Rifleman's-rule cosine for angle-corrected range — within ~1 yd for shots under 60 yd.

§ 9Sharing your builds (v135)

The Share button renders the same 1200×630 OG card composer in the Wizard Launch view, Range Log session row, and Compare overlay. Three card variants — SE-BLD-LIVE, SE-RNG-LIVE, SE-CMP-LIVE — carry the Sparrow Concierge emblem and the "powered by The Forge" pin.

What is not on a card by default: your name, initials, location, photos. The Share modal exposes opt-in toggles.

§ 10Cloud sync + aggregate data contract (v136)

The Forge is anonymous by default. Signing in is opt-in. Contributing to the community dataset is a separate opt-in. Both reversible at /pages/privacy.

The aggregate dataset (v136.1 — opt-in only). Quarterly community data cut published on the Methods page. Sign out, delete, export — full self-serve.

§ 11Spine What-If + cut-length guardrails (v135.2 / v137)

When your selected spine reads out of window, a Find your in-window spine panel renders below the verdict. The matrix calls the same canonical recommendSpineFromInputs the main verdict uses — they can never disagree at the same inputs. This is a tuning-bench what-if — the panel never mutates your saved spine.

The Recommender clamps auto-suggested cut lengths to (AMO draw − 1″) as a hard floor for rest and broadhead clearance. Override allowed but flagged with a severity-scaled SafetyNote. Source: Easton Tech Bulletin + John Dudley's School of Nock guide + K2 Arrow Co's testing program.

§ 12Where the Ashby model breaks (v140)

Dr. Ed Ashby's penetration work is a foundational contribution to bowhunting. Heavy + high FOC + clean-cutting head was a real corrective to the "fastest possible arrow" arms race of the early 2000s. The Forge uses Ashby thresholds as one input into its recommendation engine and cites his work in the build review.

But the Ashby model anchors penetration to arrow specifications — mass, FOC, broadhead type — and treats the arrow's flight quality at impact as if it's a controlled variable. In a lab gel test that's reasonable. In the field it isn't.

What the model doesn't capture, at least four variables: (1) yaw at impact — an arrow flying with 5+° tail-kick converts measurable forward momentum into shearing forces, dropping penetration 12–37% in the 5–12° range; (2) cutting-plane angle of the head to the first bone surface struck — mechanical and hybrid blades that perform beautifully on broadside soft tissue can fail to deploy or deflect at 30° impact angles; (3) speed-vs-mass tradeoff in in-flight yaw recovery — a heavy slow arrow with launch yaw has less drag-time to damp before impact; (4) mass-class-specific bone density on the largest African game — real-world performance on 1,500+ lb game (eland, large kudu, gemsbok bulls, cape buffalo) requires a steeper mass-or-tune correction than the published curve predicts.

What the Forge uses instead: Ashby thresholds as the starting frame, then four corrections derived from documented field anchors — a tune-quality penetration multiplier based on measured or estimated impact yaw, a shot scenario × head type deflection probability matrix, a flight-recovery factor that increases the penalty on lighter-faster builds at known impact yaws, and a mass-class correction on the heaviest game. The math, the calibration anchors, and the under-determined regions of the model are all published openly. We will be wrong sometimes. When we are, we update the model and publish the correction in the version history.

§ 13The Sparrow Forge Score formula (v140)

The Sparrow Forge Score is a single 0–100 number that summarizes how a build is expected to perform in real conditions. It exists because the alternative — a wall of separate numbers — leaves a thoughtful archer paralyzed and a casual one over-confident in whichever number a marketer told them to optimize.

Forge Score = 40 × tune_quality_score
            + 25 × momentum_score
            + 20 × effective_FOC_penetration_score
            + 15 × energy_retention_at_distance_score

Tune quality (40%) is the largest weight because nothing else matters if the arrow is fighting itself at impact. Measured from the Range Log's photo detection when available, estimated from slider input otherwise. The score is driven by something only Forge can measure — that's the moat.

Momentum (25%) — mass × velocity. Does the most penetration work on big game once tune is held constant. Conservative across the whole flight.

Effective FOC penetration (20%) — not raw FOC; the diminishing-returns curve that turns FOC's geometric benefit into a measurable score. Above 15% FOC the marginal penetration benefit gets small while the trajectory penalty grows.

Energy retention at distance (15%) — what the build still has at the range the user will actually shoot. Prevents the score from over-rewarding light-fast builds for hunting contexts they're not actually good at.

Tier labels: Aerospace (90–100), Surgical (75–89), Sharp (60–74), Tuned (<60). Honestly diagnostic — the score will say "Tuned" (the lowest tier) on a build that needs work. We will not call a build "Elite" because the user typed inputs in. That word has been claimed by other tools in the category for that purpose; we will not use it.

The score is open: published formula, published calibration anchors, published under-determined regions of the math. When we are wrong we will publish the correction here.

§ 14Where we punt

Reynolds-dependent C_d (under 1″ cost at 100 yd), Magnus on the shaft (self-cancels around the spin axis), Coriolis (under 0.05″ at archery range), humidity (under 0.5% air-density shift), Archer's paradox (only matters in the first 3 yd of flight), full 6-DoF (point-mass is industry standard for sight-tape), time-varying wind, temperature on carbon stiffness. These don't change the answer in any way you could detect.